Q.

For a real number y, let [y] denotes the greatest integer less than or equal to y. Let f(x)=tan(π[xπ])1+[x]2 Then

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a

f(x) is discontinuous at some x

b

f(x) is continuous at all x, but the derivative f(x) does not exist for some x

c

f(x) exists for all x

d

f(x) exist for all x, but the derivative fx0 does not exist for some x

answer is D.

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Detailed Solution

 

For all xR,x-π is integer

So,π[xπ] is an integral multiple of π

Consequently, tan(π[xπ])=0x.f(x)=0

f(x) is constant function 

f'(x)=0 f'(x) exists for all x

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